Is there a way (more efficient than the standard) to solve the [Sylvester equation][1] in the antisymmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive semidefinite, and the matrix $C$ is antisymmetric? Does this fact about $X$ follow from the statement?

Background: the matrices $C$ and $X$ are really bivectors, but I'm not sure if going the way of geometric algebra is helpful here.


  [1]: https://en.wikipedia.org/wiki/Sylvester_equation